How to make a round point or a rounder point in this case, where edges come together and join into one vertex?
Yoy cannot round 1 vertex.
You need to add more geometry.  

Nothing in the real world is perfectly pointy, so use a face, instead of one vertex.
By scaling it you can make it more or less pointy.  



You can use the subdivision modifier to smooth out the whole object.


Or use bevel to only smooth out the rim.